Page:ThomsonMagnetic1889.djvu/3

${\begin{array}{l}\xi +\xi _{0}={\dfrac {d}{dy}}(vf-ug)-{\dfrac {d}{dz}}\left(uh-\left(w-w_{0}\right)f\right),\\\\\eta +\eta _{0}={\dfrac {d}{dz}}\left(\left(w-w_{0}\right)g-vh\right)-{\dfrac {d}{dx}}\left(vf-ug\right),\\\\\zeta +\zeta _{0}={\dfrac {d}{dx}}\left(uh-\left(w-w_{0}\right)f\right)-{\dfrac {d}{dy}}\left(\left(w-w_{0}\right)g-vh\right).\end{array}}$

If the motion of the medium is irrotational, these conditions will be satisfied if we suppose that the motion of the dielectric gives rise to magnetic forces whose components $\alpha ,\beta ,\gamma$ are given by the equations

$\left.{\begin{array}{l}\alpha =4\pi \left(\left(w-w_{0}\right)g-vh\right),\\\beta =4\pi \left(uh-\left(w-w_{0}\right)f\right),\\\gamma =4\pi (vf-ug).\end{array}}\right\}$

(1)

If we suppose that the electric field is due to a number of charged spheres moving with velocities $\left(u_{1},v_{1},w_{1}\right)\ \left(u_{2},v_{2},w_{2}\right)\ \ldots$ respectively, and producing electric displacements whose components are $\left(f{}_{1},g_{1},h_{1}\right)\ \left(f_{2},g_{2},h_{2}\right)$ the component of the magnetic force parallel to $x$ will be

$4\pi \left(wg-vh-\left\{w_{1}g_{1}+w_{2}g_{2}+\ldots -v_{1}h_{1}-v_{2}h_{2}\ldots \right\}\right)$

where $f,g,h$ are the resultant displacements.

Thus, since in the general case when the æther is in motion the assumption that the currents are merely due to the changes in the polarization caused by the æther moving from a place where the displacement has one value to another where it has a different one is insufficient if the circuits are closed, it is necessary to replace it by another; the assumption we shall adopt is that the motion of the polarized æther sets up magnetic forces whose components are given by equations (1).

When the æther is at rest this agrees with Maxwell's principle that the currents are equal to the rate of increase of the electric displacement. We should get these magnetic forces if, in the expression for the mean Lagrangian function of unit volume of the moving æther, there was the term

${\begin{array}{r}a\left\{wg-vh-\sum \left(w_{1}g_{1}-v_{1}h_{1}\right)\right\}+b\left\{uh-wf-\sum \left(u_{1}h_{1}-w_{1}f_{1}\right)\right\}\\\\+c\left\{vf-ug-\sum \left(v_{1}f_{1}-u_{1}g_{1}\right)\right\},\end{array}}$

where $a,b,c$ are the components of the magnetic induction.

This term would show that there is an electromotive force parallel to $x$ equal to

$cv-bw,$